In Esnault-Viehweg's formulation, the complex structure on the underlying curve
is fixed, and the residues of the differential equation at regular singular
points are modified to achieve semistability. Flipping this around,
Biswas-Heu-Hurtubise's analogue asks if semistability can be achieved by
modifying the complex structure and fixing the residues. They claim that this
is always possible in the logarithmic, parabolic, and irregular settings, in
[1]}, [2]}, [3]}. After discovering the
Hodge-theoretic counterexample to these claims in
corollary:counterexample, we proved theorem:hn-constraints
as an attempt (1) to understand to what extent Biswas, Heu, and Hurtubise's
question:bhh has a positive answer, and (2) to apply the cases when there is a positive answer to the analysis of variations of Hodge structure on curves.
